= Solution
The pullback of the flat <Euclidean metric> under $x'{}^\mu=x^\mu+\epsilon\omega^\mu(x)$ is
$$
\delta_{\mu\nu}+\epsilon(\partial_\mu\omega_\nu+\partial_\nu\omega_\mu)+O(\epsilon^2).
$$
For a <conformal transformation> it must be a scalar multiple of $\delta_{\mu\nu}$. Taking the <trace> determines that multiple. With $\rho=\partial_\alpha\omega^\alpha$, the necessary and sufficient infinitesimal condition is the <Conformal Killing equation>
$$
\boxed{\partial_\mu\omega_\nu+\partial_\nu\omega_\mu=\frac{2\rho}{d}\delta_{\mu\nu}}.
$$
Differentiate its $(\alpha,\beta)$ instance with respect to $x^\gamma$, its $(\alpha,\gamma)$ instance with respect to $x^\beta$, and its $(\beta,\gamma)$ instance with respect to $x^\alpha$. Adding the first two and subtracting the last, commuting <partial derivatives>, leaves $2\partial_\beta\partial_\gamma\omega_\alpha$. Hence <flat conformal Killing integrability> gives
$$
\boxed{\partial_\beta\partial_\gamma\omega_\alpha=\frac1d(\delta_{\alpha\beta}\partial_\gamma\rho+\delta_{\alpha\gamma}\partial_\beta\rho-\delta_{\beta\gamma}\partial_\alpha\rho)}.
$$
Contracting $\beta$ with $\gamma$ gives $\Delta\omega_\alpha=(2-d)\partial_\alpha\rho/d$. Its <divergence> is $\Delta\rho=(2-d)\Delta\rho/d$, so $2(d-1)\Delta\rho=0$: thus $\boxed{\Delta\rho=0\text{ for }d>1}$. Taking instead the $\alpha$ <divergence> of the displayed second-derivative identity gives
$$
d\partial_\beta\partial_\gamma\rho=2\partial_\beta\partial_\gamma\rho-\delta_{\beta\gamma}\Delta\rho.
$$
Consequently $\boxed{\partial_\beta\partial_\gamma\rho=0\text{ for }d>2}$.
On a connected flat domain, write $\rho=d\lambda+2d\,b\cdot x$, with constant $\lambda$ and $b$. Integrating the second-derivative identity, and imposing the <Conformal Killing equation> on its remaining affine part, gives the complete <Conformal Killing vector field>
$$
\boxed{\omega^\mu(x)=a^\mu+R^\mu{}_{\nu}x^\nu+\lambda x^\mu+2(b\cdot x)x^\mu-b^\mu x^2},\qquad R_{\mu\nu}=-R_{\nu\mu}.
$$
Indeed the quadratic part has precisely the required <Hessian matrix>, while the symmetric part of the remaining linear term must be $\lambda\delta_{\mu\nu}$. The constants generate <translations>, <rotations>, <dilations> and <special conformal transformations>, respectively. There are $d$, $d(d-1)/2$, $1$ and $d$ independent parameters, so the full <conformal group> has dimension
$$
\boxed{\frac{(d+1)(d+2)}2}.
$$
Here “full” means local transformations, or global transformations of the <conformal compactification of Euclidean space>; its connected group is locally $SO(d+1,1)$. Requiring every transformation to be nonsingular on all of uncompactified $\mathbb R^d$ would exclude the special <conformal transformations> with finite poles.
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